Differential equation.- There are two ways to limit
the differential equation for the potential in conical
fields to small additional velocities. u, v, w, and to
limit the differential equation for nearly parallel
spatial flows to conical fields; the first is the his-
torical one, the second, however, the simpler one.
Therefore here the second one is chosen. As is well
knovtn, the linearized differential equation in the
space x, y, z, for the additional potential over a
basic velocity W in the direction of the axis z reads
if the gas has the sonic velocity a:

SI + e + z z I 0 (1)

The coordinates & and n of the conical current cor-
respond to the spatial coordinates x and y in the
plane z = 1:

: = and (2)
z z

The additional potential c increases on each ray through
the fixed point P in proportion to the distance. There-
fore, the potential divided by z is invariant on a
single ray, and provides the potential of the conical
flow:

= p(x, y,z) (5)
z

The additional velocities u, v, w are the d. rivatives
of the former potential.

(4)
= =. x- -T

The differential equation for the new potential ) is
determined from the old differential equation, and one
obtains:

NACA TM To. 1100

7 )- + 2 = 0

1 1 I
with = -
a tg a

It certainly is gratfy-ing to recognize in the type of
this dilffe.rential equation an oic acquaintance fro: the
plane gas ilow: for th; steac;.' function of the plane flow
transformi.d acco-din, to Legendre and supevimose.d over
the coi:.ponents of the current d-n.-:ity p-roc.duces exactly
the s ric d'i'frcntial equation. in rdiinar- gaz;., how-
ever, tt.e d. C-rorinatoi L is a Ical f.uncLu .on; buLt. there
is a special gas tdit-i -'rectilin-;ear a.dabatics in the
pressujle--olum-c;-diagr.i iin whi.lch,, as r-equired, thl-
denominrt,,r also re.aains cnEst&iu. Thns -as is a special
favorite w.ere it is a rmere :ues :;on of numn.-rical
calculations.

Regions of influence.- The spatial differential
equation: of the as flox at supersonic velocity i: of
hyperbolic char-acter, as shov.'n in equation (i). That
means: eash point of the flocw d-ominates a conical ran-e
opening dlownsistrea..i; eaach locus, on the other hand, is
dominated solely by those points which are situa-ced in
the cone porol-onged bacnkwvard and opening ups3trean. Ier?-e
with tlie relationships are divided definitely among the
three possibo'.lities:super-ior, subordinate, ant. independ-enI.
Machts cones in the supersonic flow cons-dei.'ed as --regiorns
of dis Ltrtance of a small trial body nmal:e this fuller coi.-
prehensible i-- a phyisal sence. It must seei d od at
first th .t th e deoondencies of the gener-al spatial flow'
are v-idelned as soon as one p'oces is to a more limited
spatial 'low. But the abcve-nentio.ned differential e _ia-
tion shows thdt inside of the circle witli the rad.is A
there rEtv_ ilr the clliptic character.

This 1beavior is easily explained by the fact that
all points of a. ray starting froi P are comiprehended
as a whole. The relation of dependencies of two rays
results from the dependencies of the single points. Only
the characteristic "indendendent"; appears uniformly in
certain cases for all -pairs of points (P itself is
excluded). The combination superior and independent

NACA TM No. 1100

becomes superior; subordinate and independent become
subordinate. But if there are pairs of points of all
kinds on the rays, then the rays are subject to the new
characteristic "reciprocally dependent." Rays of this
kind fill out the interior of I-lachts cone starting from
point P.

Characteristics.- Miacl's cone starting from P
intersects 'the ane z = 1 on the circle having the
radius A. In the field outside of this cone, i. e.,
outside of the circle in the intersecting plane, one
gets rectilinear characteristics of the differential
equation (5) which are tangents of the circle. In fig-
ure 2 this is demonstrated by two wires, a and b.
The wire b is bent slightly upstream in order not to
exclude cases of this kind. The range of disturbance
results from the sum of all of Mach's ccnes starting from
all points of the wire. It is immediately obvious that
only the circle with the radius A and its tangents can
form the boundaries of the area of disturbance. Outside
of :ach'is cone starting from point P these character-
istics settle all questions; they can be traced back to
the plane case with a transverse component of the velocity.
The essential and different part of the conical fields,
therefore, is concerned with the convex surface of ITaccts
cone starting from point P, and with its interior.

Tschapligin's illustration.- In the plane of inter-
section = 1 iv.- i_ nd .nsi,:r e f the circle with the
radius A the elliptic character of the differential
equation (5). HTear the center the differential equation
of the potential theory is valid; in plane cases, this
equation can be satisfied by analytic functions of the
complex variable. In this circle, therefore, there only
exists a mutual dependence but not yet a full equivalence
of all loci. This is not surprising, because the analytic
continuation of the plane reaches to the outer range of
the circle. Tschapligin, however, has devised a geomet-
rical construction which so distorts the field inside the
circle that equivalence regarding the differential equa-
tion will result. As figure 5 shows this distortion is
attained by transferring the plane z = 1, with the
complex variable L = + iT, through parallel projec-
tion to a sphere with the radius A, and by then pro-
jecting it from a pole of the sphere on to a plane in
the distance. One will easily recognize that only the
interior of the circle with the radius A will be.

NACA TLM No. 1100 5

depicted; first it will be delineated from the lower
half-sphere on the interior of the unit circle of the
new plan i wvith1 the new complex variable c; a second
tire it will go from the upper half-sphere on to the
outer field of the unit circle. In these coordinates
one can use analytic functions for the solutions.

SCLUTIOIN :-P T~ DIFFEREITTIAL EQUATION

For each of the velocity components u, v, and w
one can eqcuacc the real part uf an analytic function f(c).
It will serve the purpose best to set up the equation
for the component w, because tnen the more closely.,
related ccrmipnents u and v can be calculated jointiya

w = A Re (f(c)) or w + iz = A f (E) (6)

The completion represented here by s is for the time
being completely .ne aningless. According to Tschapligin
there then results the complex : velocity:

0 = u + i = I L + edf (7)
2j E

The pressure in the current, with the aid of the den-
sity p, results from the velocity components as follows:

S1.12 + v + w
P = -Pp + 2 w
w=-p "+ 2

f (E)
= ---p (f +f) +

The right function f(e) is to be selected with the aid
of the boundary conditions.

Doundary conditions.- The outside of T.ch's cone is
superior to I'[ach's cone itself. Therefore, first, those
velocities u, v, and w on the circle of the b-plane
(and therefore on the uniform circle of the e-plane)
that result from the outer field must be ascertained.

NACA T1M No. 1100

If the body does not protrude anywhere out of Machts
cone, the values u = v = w = 0 on the uniform circle
are given. If on the contrary no part of the body is
inside of Mach's cone, the values of w are to be
represented by an analytic function f(C) free of
singularities with the given boundary values on the
circle. If f(c) does not produce a stationary value
df = 0, then u and v according to equation (7) -i'll
have a logarithmic singularity at zero. The many-leaved
function can be selected in a unique way by using radial
intersections with the boundary values of u and v on
the uniform circle. The radial intersections produce
rotational layers, as is physically to be expected from
a lifting surface.

Impermeable boundaries of the body can be trans-
ferred into the E-plane at the sane time. They must be
streamlines in the field of the relative velocity:

rel = + E 0W 2A (9)

This condition is not always easy to comply with. How-
evor, if the body possesses rectilinear surface elements
passing near zero, the otherwise meaningless imaginary
part s of the function f(C) will remain constant on
these elements. If the straight part goes over zero, a
stationary value for f is to be stipulated at zero.
Conditions of this kind are especially agreeable. From
the pressure equation (8) conditions applicable to cases
of given pressures or of given lifts are to be understood.

The disappearance of the real or of the imaginary
part of f on certain lines because of symmetry can be
attained in the well known way by reflexion, as the
examples will show.

Examples

1. The circular cone in the straight flow

For the only axial-symmetrical case, i. e., the
circular cone with an infinitesimal apex angle, the
right solution is, of course, given by the statement

w + is = C Inc

NACA Ti': No. 1100

The pressure on the convex surface of the cone results
in the k:no'wn way (fig. L.) and conforms with v. Karman's
values and mine.

2. The cir1culavi cone in oblique flow

One succeeds, with the aid o?: the relative velocity
according to equation (9), in solving the circular core
in oblique flao. Herein the apex angle and the .-:ngle of
attack miy;,-, though infinitesimal, yet bear a reltiCon to
each otler. The solution is bhc.wn in figure 5. It' one
makes thr. ancle; of obliquity y scro, one gets again the
circular cone in straight flow. If' onea makes the apex
angle 2P disappear, one ge'u the pressure distribution
of a circ.ulcr cor- in an irnoj:':;i.ssible current. The
comparison v.iti] Ferrari. is rendered somewhat diffici'.lt
by the fac.t that 'Ferr ;Pari m easures Lhe v:locitv fieldI. par-
pendicular to ti.e .core axi!s wirle it is -,here perpendicular
to the wind direction. If the system of coor.dir-abes is
rotated adequately, the confom.idty is cor. plete.

5. Tip of n rectang-ular plate

If a plane ractaniguler plate of ii-'finitesimnal thick:-
ness is placed in a flo' perpendicular to the front eo.:e
with an ird;finites i1nt1 ant-le of attack, and if the velocity
field is needed only up to the rear edge of the plat-,
one can pl.ce the fi'ed point P at the right corner
point of the front edge. On the supposition of an
infinitesimal an-ls of attack y ('withi the x-ais
forming the -axis of rotation) the pressure -disZLribution
will be represented on the quarter plane between tne
positive axis:3 (z) and the negative ax:is (v). For the
plane z = 1 tih section of the body, except for infini-
tesimal distances, is then rendered by the negative real
axis. Let the reduced pressure above the plate an-i the
increased pressure below the plate be adjusted to a unit
value outside of Mach's cone. These values hold on the
boundary. circle. On the left -half of the unit circle of
the E-plane corresponding values for w arc then to be
assigned. On the rig-ht semicircle the outer field is
undisturbed; here w = 0. For reasons o.C s.yr ietry the
value w = 0 must also result on the positive real axis.
Along the negative real axis, on the contrary,
s = Im(f(e)) must be fixed because of the fixed radial
boundary. Since s is given only up to one constant,
one can demand here s = 0. All conditions can be

NACA TM No. 1100

attained by reflexion if one undertakes a preliminary
conformal mapping on the plane V = Ve.

The solution is represented in figure 7. Figure 8
shows the pressure distribution on both edges of a rec-
tangular plate.

4. Supporting triangle

Every two radii starting from P form a triangular
plane as far as the plano z = 1, when all points are
connected. Because of the required infinitestual dis-
turbance of the parallel current, however, the angle of
attack must be infinitesimal, so that the plane of the
two rays will nearly pass through the z-axis. Such
triangles are possible completely inside of ,.!a.i's cone,
completely outside, and uni- and bi-laterally protr"ding.
Here we shall only consider the simplest case of the
supporting; trian-gle outside of i.ach1s cone, although all
other cases can be easily integrated.

Figure 9 shows this supporting triangle. The
velocity component w which predominantly influences
the pressure is different from zero only on the short
arcs between t a:LId t3 as also 4 and t, The
value zero results from the undisturbed state on the
right, and also on the left because of the pressure
adjustment behind the triangle, when consideration is
given to the syrmetry with a positive and with a ne.ga-
tive angle of attack. Figure 10 shows the relations in
the t-plane. If one intends to let the rear edge *-f'
the triangle travel while the front edge lies fixed, one
will at first transfer only the points and ts into
the e-plane. With suitable regulation there must rc-ul.t
an increase of w from 0 to +TI at t, and from
-r to 0 at t2 (if one moves on the circle in the
direction of increasing angles). One can treat this
part of the solution independently if one assumes a
further singularity at zero. Physically speaking, one
then has a uniformly loaded triangle between the front
edge at am a.-i the z-axis. Inside of lac-t-s conr,
however, t is triangle is not flat, but is twisted to
uniform or loT..d. As soon as onr- supor"r-.;oaoscs at
the rear edge a negatively loaded triangle and its influ-
ence between ,3 and the z-axis, the part behind the
rear edge will no longer be supporting, and the singu-
larity in w at the point zero of the C-plane disangears.

NACA TiM No. 1100

However, a vortex layer in the field u, v is left.
The partial solution in the e-plane is represented in
figur- 11.

5. Superposition of two conical flows

The infinitesimal conical flows can be superposed
without having the fixed point in coiLrmCon as in figure 4.
Therefore the relations in the plane late can also be
represented whv.-en the pl.tte ha. mr-)ri depth. Figure 12
shov:s the isobars of the edge cC the plate and also their
superposition ~i'ter M.ach's coins 'havi overlapped. Dif-
ferent boundary .c j.itions for the partial solutions
need be cosiered osinly vwhen thL- c'oes reach the other
edge of the plate. The ,.iaa ,c, irance of ipre-uro alon.
a straight line in- just the ist tnce ai which the cones
arrive at th,; ct ..r edge of tl-e plate :,s zerm!:ablc.
The posjtivv..ly loaded prt of the p ate nids here. Fij-
ure 1i shc.o s' the 1-ft distribution of the positively
loaded part in perspective repr aesentation.

To find the velocity field behind a. rectcar-ulaL-
plate of finite dcpth one can E:en.pul the supporting; : pres-
sure differiEinces of a pletas of infinite' 'crith by con,ical
fields having t the apices on the r:ar cdgc of the plate.
If one superposes a negatively. loaded ol.ate shifted
infinritosimall- in the di'rcotion of the z-a
obtains a su-pporting line as a,: limiting case of the
supporting sirip. The cases calculated by S-i-lichting
according to Prandtl's :,imetliod a e obtained in this way.
Here, too, the conformitv is perfict, except for an
error of in in thealculatn these Integral equation.

SUMNl WRY

The calculatlon of irfinitesimal conical supuersc.nic
flows has been applied firvt to the simplest xmir.,lcs
that have -iso been calculated in anotlt.her way. Except
for the discovery of a miscalculation in an older rencrt
there was found the expected conforriity. The taew Tmithod
of calcullation is limited more definitsi.y to the conical
case." tub, as a compensation, it is much more convenient
because tlh solution is obtained by analytic functions.
The fundamental recognition that; there the hyperbolic
character is replaced by the elliptic one will lead to

NACA Ti No. 1100

more thorough investigation of conical fields as special
cases in supersonic flows. Of course, one will be
tempted to call the elliptic character seemingly elliptic
only; for if one notches an indentation into a cone, the
real hi-. rbdlic character of the field of the flow down-
stream from this indentation immediately becomes obvious
again. However, if one traces the flowr to a point very
far behind the indentation there will appear inside of
liacl- ts cone a reciprocal relation between every two rays
which will gradually restore the donical course of the
flow. Instead of trying to produce the final flovw by an
infinite succession of hyperbolic dependencies it will
be more expedient to consider special elliptic singu-
larities at the points of disturbance. In those rela-
tions I see the significance of the conical field; the
infinitesimal case represents only a first approximation
to it.

Translation by :,.-y L. :ibler
and Robert T. Jones,
National Advissory Committoee
for Aeronautics.

NACA TM No. 1100

P

Figure 1. Coordinates in conical field

Figure 2. Disturbance field of elements a and b

Figs. 1,2

NACA TM No. 1100

Figure 3. Chaplygin's transformation

W 'gg L? f d

f = Clne

A = tg
A = tgQ:

Figure 4. Circular cone in axial flow.

Figs. 3,4

A 2 (EI +
T= z~e

NACA TMNo. 1100 Figs. 5,6

y

I -
p o

f = 4WAR2

k n

p =Qe ,2 (8 I

S=-- ita

1
1--
C-E
0 E

2A _
S-- -
r,
/3 2

.0
oE

1-EE a

2
r-o

y- + 2 ycos i+Y- r0c 2)

Figure 5. Circular cone in yawed flow

W

1 apr. tga.y tga-eatt
- ( d-#s) = arc cos
2 77 tga

A-.tfi

Figure 6. Edge of a rectangular Tlate

-2 :
+ '0
l-E60)21

Figs. 7,8,9

Figure 7.

NACA TM No. 1100

v = r-,

f(v) =--i { In (v- ) -In(v-v2)

-In(v-v3)+ In(v-v4)

Conformal representation at edge of a
rectangular plate

Direction of motion of
plate

Figure 8. Pressure distribution on a flat plate

Figure 9. The
lifting triangle.

NACA TM No. 1100

2Ae
1+e"

Figure 10. Cross section of the lifting triangle

ffe) = i {n(e-2)+tin(-e)-ine

pd-ps = const.

Figure 11. Representation of the lifting triangle
in the e plane

Figs. 10,11

NACA TM No. 1100

Figure 12. Superposition of edge influences for
the rectangular plate at supersonic velocities